Research Article Open Access

Li and Wang, J Biomet Biostat 2012, S7http://dx.doi.org/10.4172/2155-6180.S7-007

Research Article Open Access

Biometrics & Biostatistics

J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journalMedical statistics: Clinical and Experimental Research

Keywords: Influence function; Instrumental variable; Longitudinal data; Measurement error; Mixed effects; Outliers; Robustness; Simulation-based estimator

IntroductionGeneralized linear mixed models (GLMM) have been widely used

in modeling longitudinal data where the response can be either discrete or continuous. Various estimation methods for GLMM have been developed in the literature [1,2,3]. However, estimation and inference in a GLMM remain very challenging when some of the covariates are not directly observed but are measured with error.

Measurement error is a common problem in medical and clinical research. For example, in epidemiologic studies time-varying covariates such as dietary intakes or other exposure variables are subject to measurement error; in HIV trials, the viral load and CD4+ cell counts are often mismeasured. It is well-known that simply substituting a proxy variable for the unobserved covariate in the model will generally lead to biased and inconsistent estimates of regression coefficients and variance components [4-6]. To account for the measurement error as well as the correlation in the longitudinal data, Wang et al. [5] proposed the simulation extrapolation (SIMEX) method to correct for the bias of the naive penalized quasi-likelihood estimator in a generalized linear mixed model with measurement error (GLMMeM), while Wang et al. [7], and Bartlett et al. [8] proposed a regression calibration (RC) approach. Although both RC and SIMEX approaches are practical and may produce satisfactory results when the measurement errors are small, it is known that in general they yield approximate but inconsistent estimators. Tosteson et al. [9] proposed a bias-corrected estimator but it was shown to be highly inefficient. Buonaccorsi et al. [10] proposed the likelihood based methods and Zhong et al. [11] studied the corrected score approach. However, the likelihood methods typically entail computation difficulties due to the multiple integrals and intractability of the likelihood function. Consequently, one usually relies on normality assumption for random effects, measurement error and residual error terms. Non- or semi-parametric approaches have also been considered for models with normal ME [12,13].

Instrumental variable (IV) method has been used by many researchers to overcome measurement error problems in cross-sectional data [6,14-17]. In practice, any variables correlated with the error-prone true covariates can serve as valid IV, e.g., a second independently measurement. Furthermore, the assumption of

instrumental variable is weaker than that of replicate data because IV can be a biased observation for the true covariates [6,18].

In this paper, we propose an exact consistent estimation method for GLMMeM based on the methods of moments and instrumental variables. This method is easy to implement when the closed form of the moments exist. For the case where the marginal moments do not admit analytic forms, we propose simulation-based estimators (SBE). In particular, we use the simulation-by-parts technique of Wang [19] to construct the SBE to ensure its consistency even using finite number of simulated random points. The proposed estimators are root-n consistent and do not require the parametric assumptions for the distributions of the unobserved covariates or of the measurement errors. Further, the proposed estimators have bounded influence functions so they are robust to data outliers.

The structure of the paper is as follows. In Section 2, we introduce the model and the proposed moments estimator. In Section 3, we construct the simulation-based estimator. In Section 4 we present simulation studies of finite sample performances of the proposed estimators. Finally, conclusions and discussion are contained in Section 5, whereas theoretical proofs of the theorems are given in Section 6.

The Model and EstimationConsider the following generalized linear mixed model with

measurement error (GLMMeM)

( )1 ( | , ) = ' ' ' , = 1, , , = 1, ,β β− + + ij i ij ij x ij z ij i ig E y b X X Z B b i N j n (1)

( | , ) = ( ( ' ' ' ))φν β β+ +ij i ij ij x ij z ij iV y b X g X Z B b (2)

Where yij ∈ IR is the jth response for the ith unit; bi ∈ IRq is the random effect having mean zero and distribution fb (t; θ) with unknown parameters q ∈ pbIR ; β ∈ px

x IR and β ∈ pzz IR are vectors of

*Corresponding author: Liqun Wang, Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2, Tel: (204) 474-6270; Fax: (204) 474-7621; E-mail: liqun_wang@umanitoba.ca

Received March 20, 2012; Accepted April 12, 2012; Published April 13, 2012

Citation: Li H, Wang L (2012) Consistent Estimation in Generalized Linear Mixed Models with Measurement Error. J Biomet Biostat S7:007. doi:10.4172/2155-6180.S7-007

Copyright: © 2012 Li H, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Consistent Estimation in Generalized Linear Mixed Models with Measurement ErrorHe Li and Liqun Wang*Department of Statistics, University of Manitoba, Canada

AbstractWe propose the instrumental variable method for consistent estimation of generalized linear mixed models with

measurement error. This method does not require parametric assumptions for the distributions of the unobserved covariates or of the measurement errors, and it allows random effects to have any parametric distributions (not necessarily normal). We also propose simulation-based estimators for the situation where the marginal moments do not have closed forms. The proposed estimators are not only computationally attractive but also strongly root-n consistent. Moreover, the proposed estimators have a bounded influence function so they are robust against data outliers. The methodology is illustrated through simulation studies.

Citation: Li H, Wang L (2012) Consistent Estimation in Generalized Linear Mixed Models with Measurement Error. J Biomet Biostat S7:007. doi:10.4172/2155-6180.S7-007

Page 2 of 7

J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journalMedical statistics: Clinical and Experimental Research

( ( ; ) ) ' ' ( ; ) ( ; ) ,ϕ φ ν g β β q α′ + + + ∫jk ij x ij z ij b Ug m V u Z B t f t f u dtdu

and

k3,ijk(ψ) = E(yijWik | Vi) (8)

= E(yijXik | Vi)

= E [XikE(yij | bi, Xi) | Vi]

= ( ( ; ) ) ( ( ; ) ) ' ' ( ; ) ( ; ) ,g g β β q α′ + + + + ∫ ik ij x ij z ij b Um V u g m V u Z B t f t f u dtdu

where φjk = 1 if j = k and zero otherwise. In the following we consider three popular GLMMeM examples.

Example 2.1: Consider a linear mixed model with continuous responses and an identity link function g (-). Assuming Uij has mean zero and variance matrix αI, and bi has mean zero and covariance matrix Σb, we have the explicit form of the moments

1, ( ) = ( ; ) ' ,κ ψ g β β′ +ij ij x ij zm V Z

2, 1, 1,( ) = ( ) ( ) ,κ ψ κ ψ κ ψ ϕ αβ β ϕ φ′ ′+ Σ + +ijk ij ik ij b ik jk x x jkB B

k3,ijk(ψ) = k1,ij (ψ)m(Vik;γ) + φjkαβx

It is worth noting that no distributional assumptions are required for Uij and bi to obtain these moments.

Example 2.2: Consider a random intercept mixed Poisson model for counts, where logE(yij|bi,Zi,Xi,Bi) = β0 + βxxij +βzzi + βxzxijzi +bi and ϕ = 1; xij, zi and bi are scalars. Assuming bi : N(0, q) and uij : N(0,αI), we can derive the explicit forms of the moments as

( )2 2 21, 0( ) = exp ( ) ( ; ) ( ) / 2 / 2 ,κ ψ β β β g β β α β q+ + + + + +ij x xz i ij x xz i z iz m v z z

( )2 2 22, 1, 1, 1,( ) = ( ) ( ) exp ( ) ( ),κ ψ κ ψ κ ψ β β α q ϕ κ ψ+ + +ijk ij ik x xz i jk ijz

3, 1, 1,( ) = ( ; ) ( ) ( ) ( ).κ ψ g κ ψ ϕ β β ακ ψ+ +ijk ik ij jk x xz i ijm v z

Example 2.3: Consider a mixed logistic model for a binary response yij, where ϕ = 1 and g (-) is the logistic distribution function. For this model we find

( )1, ( ) = ( ; ) ' ' ( ; ) ( ; ) ,κ ψ g β β q α′ + +∫ij ij x ij z ij b Ug m V Z B t f t f u dtdu

( )2, ( ) = ( ; ) ' 'κ ψ g β β′ + +∫ijk ij x ij z ijg m V Z B t

( )( ; ) ' ' ( ; ) ( ; ) ,g β β q α′ + +ik x ik z ik b Ug m V Z B t f t f u dtdu

3, ( ) = ( ( ; ) )κ ψ g +∫ijk ikm V u

( )( ( ; ) ) ' ' ( ; ) ( ; ) .g β β q α′+ + +ij x ij z ij b Ug m V u Z B t f t f u dtdu

The above integrals are intractable but can be approximated using Monte Carlo simulators. This case will be treated in the next section.

Since γ is of secondary interest, it is treated as a nuisance parameter and is estimated by nonlinear least squares (NLS) method based on equation (5) as

=1

ˆ = ( ) = '( ) ( ),a ag gg g

g g g g∈Ω ∈Ω

Ψ ∑N

N N i ii

r rrgmin rgmin (9)

where '( ) = ( ( ; ),1 )g g− ≤ ≤i ij ij ir W m V j n . Under standard regularity

conditions, 1/20ˆ = ( ).g g −−N pO N Then we replace γ in (6)-(8) by its least

squares estimator g N and denote the moments as 1, 2,ˆ ˆ, ,κ κij ijk and 3,κ ijk

3,κ ijk correspondingly. Throughout the paper, we denote the parameter space of a parameter vector, say ψ, by Ωψ. In particular, the parameter spaces of βx and βz are denoted as Ωx and Ωz respectively. Then the

fixed effects; g-1(-) is a link function; v(-) is a known variance function and ϕ ∈ IR is a scalar parameter that may be known or unknown. It is assumed that yij given bi are independent. Note that the estimation methods in this paper do not require the conditional distribution of

yij given bi to belong to an exponential family. Further, ∈ pzijZ IR and

∈ qijB IR are known predictors observed without error; and ∈ px

ijX IR is unobservable. Instead one observes

Wij = Xij + dij, (3)

where dij is the vector of measurement errors.

Model (1) - (2) has been studied by various authors, e.g. Wang et al. [5]; Buonaccorsi et al. [10]; Zhong et al. [11]; Carroll et al. [6]. It is known that the parameters of the classical ME models generally require extra information in order to be identified [6,16]. Moreover, even if certain ME models are identifiable, additional information is useful to improve the efficiency of estimation [20]. The common source of additional data includes replicate measurements, validation data, instrumental variables, or knowledge of the measurement error distributions. Here we assume that one observes a set of instrumental variables ∈ pV

ijV IR that is related to the error-prone predictor Xij through

Xij = m(Vij ; g) + Uij, (4)

Where m(-) is a known function, g ∈ pvIR is a vector of unknown parameters, ∈ px

ijU IR is independent of Vij and has mean zero and distribution fU (u;α) with unknown parameters α ∈ puIR . Further, we assume that the ME δij is independent of Xij, Vij and yij, E(yij | Xij,bi) = E(yij | Xij, Vij, bi) and E(yij yik | Xij,bi) = E(yij yik | Xij, Vij, bi) where j ≤ k. Following the convention of mixed modeling literature, throughout this paper all expectations are taken conditional on Bi and Zi implicitly. There are no assumption on the functional forms of the distributions of Xij and δij. In this model, the observed variables are ( , ', ', ', ')′ij ij ij ij ijy W V Z B and the parameter of interest is = ( , , , , )ψ β β q α φ′ ′ ′ ′ ′x z .

To estimate all unknown parameters in the model, we first note that substituting (4) into (3) results in a usual regression equation

E(Wij | Vij) = m(Vij; γ) (5)

which can be used to obtain consistent estimator for γ by least squares method. In practice, γ can be pre-estimated using an external sample or a subset of the main sample. We denote 1 2= ( , , , )′ ′ ′ ′

i i i iniX X X X , and

denote Wi, Vi, Zi, Bi and Yi analogously. By model assumptions and the law of iterated expectation, we have the following moments

k1,ij(ψ) = E(yij |Vi) (6)

=E [ E(yij | bi, Xi, Vi) | Vi]

=E [ E(yij | bi, Xi) | Vi]

= ( ' ' ' ) |β β + + ij x ij z ij i iE g X Z B b V

= ( ( ; ) ) ' ' ( ; ) ( ; ) ,g β β q α′ + + + ∫ ij x ij z ij b Ug m V u Z B t f t f u dtdu

and, similarly,

k2,ijk(ψ) = E(yijyik | Vi) (7)

=E [ E(yij | bi, Xi,) E | E(yik | bi, Xi) | Vi]

= ( ( ; ) ) ' 'g β β′ + + + × ∫ ij x ij z ijg m V u Z B t

[ ]( ( ; ) ) ' ' ( ; ) ( ; )g β β q α′+ + + +ik x ik z ik b Ug m V u Z B t f t f u dtdu

Citation: Li H, Wang L (2012) Consistent Estimation in Generalized Linear Mixed Models with Measurement Error. J Biomet Biostat S7:007. doi:10.4172/2155-6180.S7-007

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J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journalMedical statistics: Clinical and Experimental Research

method of moments estimator (MME) for ψ is defined as

=1

ˆ ˆ ˆ= ( ) = ( ) ( ),a aψ ψψ ψ

ψ ψ ρ ψ ρ ψ′∈Ω ∈Ω

∑N

N N i i ii

Q Argmin rgmin (10)

where 1, 2, 3,ˆ ˆ ˆ ˆ( ) = ( ( ), ( ), ( ))ρ ψ κ ψ κ ψ κ ψ′ − − −i ij ij ij ik ijk ij ik ijky y y y W and Ai = A(Vi) is a nonnegative definite matrix that may depend on Vi.

To derive the consistency and asymptotic normality of ψ N , we make the following assumptions.

Assumption 1: g (-) and v (-) are continuously differentiable; m(v; -) is a Lebegue measurable function of v and is continuously differentiable with respect to γ.

Assumption 2: (yi,Wi,Vi,Zi,Bi,ni), i=1,…,N, are in-

dependent and identically distributed and satisfy

( )24 1 < + + ∞ i ij ij ijE A y y W ; Further, there exists a positive func-

tion G(v,t,u) satisfying ( )2( , , )( ( , ) 1) <g + + ∞ ∫E A G V t u m V u dtdu

, such that 2 ( ( , ) ) ' ' ( ; ) ( ; )g β β q α′ + + + x ij z ij b Ug m v u Z B t f t f u and

( ( , ) ) ' ' ( ; ) ( ; )ν g β β q α′ + + + x ij z ij b Ug m v u Z B t f t f u are bounded by G(v,t,u).

Assumption 3: The parameter space is compact.

Assumption 4: 0 0[ ( ) ( )] [ ( ) ( )] = 0ρ ψ ρ ψ ρ ψ ρ ψ′− −i i i i iE A if and only if ψ = ψ0.

Assumption 5: g (-)and v (-) are twice continuously differentiable; fb(t; θ) and fU(u;α) are twice continuously differentiable w.r.t to θ and α respectively in some open subsets θ0∈ Ωθ0⊂ Ωθ and α∈ Ωα0∈ Ωα. Furthermore, all first and second order partial derivatives of ( ( , ) ) ' ' ( ; ) ( ; )g β β q α′ + + + ij x ij z ij b Ug m V u Z B t f t f u and

( ( , ) ) ' ' ( ; ) ( ; )ν g β β q α′ + + + ij x ij z ij b Ug m V u Z B t f t f u w.r.t (ψ,γ) are bounded absolutely by the positive function G(v,t,u) given in Assumption 2.

Assumption 6: The matrices

0 0( ) ( )= ,ψρ ψ ρ ψ

ψ ψ′∂ ∂

′∂ ∂ i i

iD E A (11)

0 0( ) ( )=gg gg g

′∂ ∂ ′∂ ∂

i ir rD E (12)

are nonsingular.

Theorem 1: As N→∞,

1. under assumptions 1-4, . .

0ψ ψ→a s

N ;

2. under assumptions 1-6, 1 10ˆ( ) (0, )ψ ψψ ψ − −− →

L

NN N D CD , where

C =E(C1C1) (13)

101 0

( ) ( )= ( ) ( )ψg gρ ψ gρ ψ g

ψ g−′ ′∂ ∂

+∂ ∂i i

i i irC A D D r (14)

0 0( ) ( )=ψgρ ψ ρ ψ

ψ g′∂ ∂

′∂ ∂ i i

iD E A (15)

The second term in equation (14) is the correction term due to the first-step estimation of γ. If γ0 is known or estimated using an independent sample from the main sample, then this term vanishes and the most efficient weight is given by 1

0 0= [ ( ) ( ) | ]ρ ψ ρ ψ −′

opti i i iA E V

[21]. In practice, direct calculation of optiA is not feasible since it

involves unknown parameters to be estimated. One possible solution is using a two-stage procedure. First, minimize ( )ψNQ using Ai = I to obtain the first stage estimator 1ψ N . Second, estimate opt

iA by any nonparametric method or

1

1 1=1

1 ˆ ˆ= ( ) ' ( ) ,ρ ψ ρ ψ−

∑N

opti i N i N

iA

N (16)

and minimizing ( )ψNQ again using optiA to obtain the second stage

estimator 2ψ N . In practice, the calculation of optiA may be difficult

or inaccurate due to its high dimension, so one may consider using certain diagonal weight matrix. A detailed discussion on the choice of

optiA can be found in Li and Wang (2011).

In general, MME can be computed using Newton-Raphson algorithm as

12 ( ) ( )( 1) ( ) ˆ ˆ( ) ( )ˆ ˆ= ,

τ ττ τ ψ ψψ ψ

ψ ψ ψ

−

+ ∂ ∂− ′∂ ∂ ∂

N NQ Q

where ( )ˆ τψ denotes the estimate of ψ at the τth iteration, ( ) ( )

( )

=1

ˆ ˆ( ) ( ) ˆ= 2 ( ),τ τ

τψ ρ ψ ρ ψψ ψ

′∂ ∂∂ ∂∑

NN i

i ii

Q A (17)

2 ( ) ( ) ( ) ( )

=1

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ( ) / )ˆ= 2 ( ( ) ) .τ τ τ τ

τψ ρ ψ ρ ψ ρ ψ ψρ ψψ ψ ψ ψ ψ

′ ′′

∂ ∂ ∂ ∂ ∂ ∂+ ⊗ ′ ′ ′∂ ∂ ∂ ∂ ∂

∑N

N i i ii i i

i

Q vecA A I

(18)

Since the second term in (18) has expectation zero, it can be ignored for computational simplicity.

When using the weight (16), the MME is able to safeguard against influential measurements. In particular, the influence function (IF) at a single contaminated data point v for subject l takes the form

1 ˆ ˆ( ; ( ))ˆ ˆ ˆ ˆ ˆ( ; , ) = ( ( )) ( ; ( )) ( ; ( )),ψρ ψψ ψ ψ ρ ψ

ψ− ′∂

−∂

l NN N l N l N

v FIF v F D F A v F v F (19)

where F is the underlying distribution and Dψ is given in (11). If ψ N is computed using the estimated weight (16), then analogous to Li and Wang [22] we can prove that ˆ( ; , ) 0ψ →NIF v F as → ∞v . Therefore, the influence function of ψ N is bounded and ψ N has a redescending property, so it is robust to influential observations or outliers.

Simulation-Based Estimator The numerical computation of MME ψ N is straightforward if the

moments in (6)-(8) admit explicit forms. However, sometimes the integrals involved in these moments are intractable. In this case, we propose a simulation-based approach. The basic idea is to replace the integrals with their Monte Carlo simulators as follows. First, generate random points tis and uis,i = 1,2,…,N; s = 1,2,…,2S; from known densities l(t) and h(u). Then use the first half of the points tis and uis,s = 1,2,…,S to compute

11,

=1

( ; ) ( ; )1( ) = ( ( ; ) ) ' '( ) ( )q ακ ψ g β β′ + + + ∑

Sb is U is

ij ij is x ij z ij iss is is

f t f ug m V u Z B tS l t h u

(20)12,

=1

1( ) = ( ( ; ) ) ' 'κ ψ g β β′ + + + ∑S

ijk ij is x ij z ij iss

g m V u Z B tS

[ ] ( ; ) ( ; )( ( ; ) ) ' '( ) ( )q αg β β′+ + + b is U is

ik is x ik z ik isis is

f t f ug m V u Z B tl t h u

( )=1

( ; ) ( ; )1 ( ( ; ) ) ' '( ) ( )q αϕ φ ν g β β′ + + + + ∑

Sb is U is

jk ij is x ij z ij iss is is

f t f ug m V u Z B tS l t h u

(21)13,

=1

1( ) = ( ( ; ) )κ ψ g +∑S

ijk ik iss

m V u gS

( ; ) ( ; )( ( ; ) ) ' '

( ) ( )q αg β β′ + + +

b is U isij is x ij z ij is

is is

f t f um V u Z B tl t h u

(22)

and similarly use the second half of the points tis and uis,s = S+1,S+2,…

Citation: Li H, Wang L (2012) Consistent Estimation in Generalized Linear Mixed Models with Measurement Error. J Biomet Biostat S7:007. doi:10.4172/2155-6180.S7-007

Page 4 of 7

J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journalMedical statistics: Clinical and Experimental Research

,2S and to compute 21, ( )κ ψij , 2

2, ( )κ ψijk and 23, ( )κ ψijk . It is easy to see

that 1, ( )ικ ψij , 2, ( )ικ ψijk and 3, ( )ικ ψijk , ι = 1,2 are unbiased estimators for 1, ( )κ ψij , 2, ( )κ ψijk and 3, ( )κ ψijk respectively. Finally, the simulation-based

estimator (SBE) for ψ is defined as

, , ,1 ,2=1

ˆ ˆ ˆ= ( ) = '( ) ( ),a aψ ψψ ψ

ψ ψ ρ ψ ρ ψ∈Ω ∈Ω

∑N

N S N S i i ii

Q Argmin rgmin (23)

where , 1, 2, 3,ˆ ˆ ˆ ˆ( ) = ( ( ), ( ), ( ))ι ι ιιρ ψ κ ψ κ ψ κ ψ ′− − −i ij ij ij ik ijk ij ik ijky y y y W . We re-

fer this simulation technique as simulation-by-parts since ,1ˆ ( )ρ ψi and ,2ˆ ( )ρ ψi are constructed by using two independent sets of random

points. The benefit of simulation by parts is that ,1ˆ ( )ρ ψi and ,2ˆ ( )ρ ψi are conditionally independent given (Yi,Wi,Vi,Zi,Bi) so that , ( )ψN SQ is an unbiased simulator for ( )ψNQ for finite S. It is worth noting that the construction of simulated moments only requires bi and Uij to have certain known parametric forms (not necessary normal). For example, one can follow Davidian and Gallant [23] and Zhang and Davidian [24] to represent the density of bi and Uij by the standard seminonparamet-ric density which includes normal, skewed, multi-modal, fat- or thin-tailed densities. One can also impose the Tukey (g,h) family distribu-tion [25] for bi as well which is generated by a single transformation of the standard normal and covers a variety of distributions.

Theorem 2: Suppose that Supp(l) ⊇ Supp(fb(.; θ)) for all θ ∈ Ωθ0, and Supp(h) ⊇ Supp(fU(.; α)) for all α ∈ Ωα0. Then for any fixed S > 0, as N→∞,

1. under assumptions 1-4, . .

, 0ψ ψ→a s

N S ;

2. under assumptions 1-6, 1 1, 0ˆ( ) (0, )ψ ψψ ψ − −− →

L

N S SN N D C D , where

( )1 1= ,′S S SC E C C (24),1 0 ,2 0 1

1 ,2 0 ,1 0

'( ) '( ) ( )2 = ( ) ( ) 2 ( )ψg g

ρ ψ ρ ψ gρ ψ ρ ψ gψ ψ g

− ′∂ ∂ ∂+ +

∂ ∂ ∂i i i

S i i i i irC A A D D r (25)

Note that the above asymptotic results do not require the simulation size S tends to infinity because we use the simulation-by-parts technique to approximate moments. This is fundamentally different from other simulation-based methods in the literature which typically require S goes to infinity to obtain consistent estimators. However, due to the approximation of marginal moments, ,ψ N S is generally less efficient than ψ N . In general, analogous to the Corollary 4 in Wang [19] we can show that the efficiency loss caused by simulation decreases at the rate O (1/S).

Monte Carlo Simulation StudiesIn this section, we evaluate the finite sample behavior of the

proposed estimators, and compare them with the naive maximum likelihood estimates that ignores measurement error. We carried out Monte Carlo replications in each simulation study and reported the biases and the root mean square errors (RMSE). All computations are done in R [26] and the naive ML estimates are obtained from glmmPQL package.

In the first simulation study, we considered the mixed Poisson model in Example 2.2. We simulated δij from N(0,1), set zi = 1 for half the sample and 0 for the remainder, and set N = 100,300 and n = 4. In addition, we set xij = 1.5 + 0.5vij + uij, vij : N(0,1), and uij : N(0,0.25). Table 1 reports the simulation results. For the fixed effects associated with ME,β0, βx and βxz, the MME is almost unbiased while the naive MLE is severely downward biased and attenuated towards zero. The MME is considerably more efficient than the naive MLE in terms of smaller RMSE. With the increase of sample size from N = 100 to 300, the RMSE and biases of MME are decreasing while the ones from the naive MLE stay almost the same. For exactly measured effect βz, the MME

still provides a better estimate in terms of biases and RMSE which may because zi interacts with xij. The naive MLE for βz is also biased towards zero. However, with the increase of sample sizes, the biases and RMSE reduces for the naive MLE as well as the MME. For the random effect θ, surprisingly both estimators provide quite satisfactory estimators with no apparent biases.

In the second simulation study, we considered a logistic model for binary responses. In particular, we adopted the following model used in the simulation studies by Wang et al. [5]:

logit(Pr(yij = 1| bi,xij, zij)) = β0 + βxxij + βzzij + bi (26)

where bi : N(0,0.5), zij : N(0,1), and δij : N(0,1). In addition, we assumed an IV variable is observed that relates to xij though xij = 1.5 + 0.5vij +uij, vij : N(0,1) and uij : N(0,0.5). In the present simulation, we selected N = 50,100 and n = 3. The closed form of the marginal moments are not available so we applied the SBE in this case. To compute the SBE, we chose the density of N(0,2) to be h(u) and l(t), and generated independent points uis and tis, s = 1,…,2S, using S = 1000. The simulation results are presented in Table 2. For the fixed effects associated with ME, β0 and βx, SBE is almost unbiased while the naive ML is severely downward biased and attenuated towards zero. With the increase of sample size from N = 50 to 100, the RMSE and biases of MME are decreasing while the ones from the naive ML stay almost the same. This is the same findings as the ones in the first simulation study. For exactly measured effect βz, both estimates seem to be unbiased; however, the naive ML provides a better estimates in terms of smaller biases and RMSE. With the increase of sample size, the RMSE and biases from both methods are decreasing. For the random effect, the naive ML overestimates θ with larger biases as well as RMSE. With the increase of sample size, both estimators lead to smaller biases and RMSE.

In the third simulation study, we considered the linear mixed model in Example 2.1 with N = 100,300 and n = 4. We simulated ϕ,vij, and uij independently from a standard normal distribution and set xij= 0.2+1.4 vij+ uij. The random effect was generated from either a normal distribution or a t(3) distribution. Table 3 reports the simulation results. For β0 and βx, the MME is almost unbiased while the naive MLE is severely biased. The MME has much smaller RMSE than MLE. With the increase of sample size from N = 100 to 300 to, the RMSE and biases of MME are decreasing while the ones from the naive MLE stay almost the same. The performance of MME and MLE for fixed effects under normal random effects and non-normal random effects are very similar. For the random effect θ, both estimators provide quite satisfactory estimators with no apparent biases. However, the RMSE from naive MLE increased significantly when the random effects were shifted from normal to t(3). For the residual error variance ϕ, naive MLE produces huge bias while MME stays unbias. These findings demonstrated that MME is robust against random effects misspecification.

N = 100 N = 300 Parameter Naive MLE MME Naive MLE MME β0 = 1.00 1.37 (1.39) -0.22 (0.27) 1.38 (1.38) -0.20 (0.23) βx = 1.00 -0.79 (0.79) -0.05 (0.08) -0.79 (0.79) -0.07 (0.08) βz = -0.50 0.38 (0.47) 0.06 (0.17) 0.36 (0.40) 0.05 (0.16) βxz = 0.25 -0.20 (0.22) -0.01 (0.10) -0.19 (0.20) -0.02 (0.08) θ = 1.00 -0.03 (0.21) -0.04 (0.19) 0.03 (0.28) -0.05 (0.15)

Table 1: Biases(RMSE) for the parameter estimates in the random intercept Pois-son models.

Citation: Li H, Wang L (2012) Consistent Estimation in Generalized Linear Mixed Models with Measurement Error. J Biomet Biostat S7:007. doi:10.4172/2155-6180.S7-007

Page 5 of 7

J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journalMedical statistics: Clinical and Experimental Research

Concluding RemarksWe proposed consistent estimators for the generalized linear mixed

models with errors in variables. These estimators are constructed by combining the method of instrumental variables and the method of moments. The strong consistency and asymptotic normality of the estimators are obtained under mild regularity conditions. The proposed approach does not require the parametric assumptions for the distributions of the unobserved covariates or the normality assumption for the random effects. In the simulation studies the proposed estimators perform well in finite sample situations.

In comparison with the regression calibration and simulation extrapolation [5,7], the proposed method is exactly consistent and computationally less intensive. It is therefore more reliable even if the measurement errors are large. Moreover, the instrumental data is usually less expensive to collect than validation data which is generally required by the regression calibration method. Unlike the likelihood-based method [10], the proposed method does not rely on normality distributional assumptions on the random effects or unobserved covariates and is easier to compute. As noted in Carroll and Stefanski [18], the assumption for an instrumental variable is weaker than a replicate for the error-prone true variable. In general, if the assumption of replicate cannot be verified, one may wish to investigate the applicability of using it as an instrumental variable. Therefore, the proposed method is less restrictive than the ones relying on replicates. A special case of measurement error is that the true unobserved covariate is known to be time-invariant (i.e., Xij = Xi) but its surrogate Wij is measured over time. In this case, Wij can be used as the instruments for Xi and hence no extra data is required for the model identification and estimation. The proposed estimators can also be extended to the case of the generalized linear mixed models with Berkson measurement error. However, the use of instrumental variable is not necessary because the model is identifiable by using the first two moments (i.e., equation (6) and equation (7)). The proof is straightforward by following Wang [27].

Theoretical proofs

Proof of Theorem 1.1

By assumption 1 and the Dominated Convergence Theorem

(DCT), we have the first-order Taylor expansion about γ0.

0=1 =1

( , ) ˆ( ) = ( ) ( ) 2 ( , ) ( ),ρ ψ gψ ρ ψ ρ ψ ρ ψ g g gg′ ′

∂+ −

′∂∑ ∑

N Ni

N i i i i i Ni i

Q A A (27)

where 0 0ˆg g g g− ≤ − N . Further, for any 1≤ ≤i N , by assumptions 1-3

and Cauchy-Schwartz inequality, we have 22 2 2 2( ) 2 2 2ρ ψ

≤ ≤

≤ + +∑ ∑ ∑i ij ij ik ij ikj j k j k

y y y y W

( )22 ( ( ; ) ) ' ' ( ; ) ( ; )g β β q α′ + + + + ∑ ∫ ij x ij z ij b U

jg m V u Z B t f t f u dtdu

(4 ( ( ; ) ) ' 'g β β≤

′ + + + + ∑ ∫ ij x ij z ijj k

g m V u Z B t

[ ] )2( ( ; ) ) ' ' ( ; ) ( ; )g β β q α′+ + +ik x ik z ik b Ug m V u Z B t f t f u dtdu

( )224 ( ( ; ) ) ' ' ( ; ) ( ; )φ ν g β β q α′ + + + + ∑ ∫ ij x ij z ij b U

jg m V u Z B t f t f u dtdu

( )22 ( ; ) ( ( ; ) ) ' ' ( ; ) ( ; )g g β β q α

≤

′ + + + + + ∑ ∫ ik ij x ij z ij b Uj k

m V u g m V u Z B t f t f u dtdu

and therefore 2' ( ) ( ) ( )sup sup

ψ ψ

ρ ψ ρ ψ ρ ψΩ Ω

≤i i i i iE A E A

( )22 2 22 ( 1)≤ + + +i i ij i i i ij ik i ij ikn E A y n n E A y y E A y W

( )2 ( , , , )+ ∫i i i in E A G V Z t u dtdu

( )222 1 2 ( , , , )sup

φ

φΩ

+ + +

∫i i i i in n E A G V Z t u dtdu

( )2

0( 1) ( , , , ) ( ; )g+ + +∫i i i i in n E A G V Z t u m V u dtdu

< .∞

Hence by the uniform law of large numbers (ULLN), . .

=1

1 ( ) ( ) ( ) 0,supψ ψ

ρ ψ ρ ψ ψ′∈Ω

− →∑N a s

i i ii

A QN (28)

where ( ) = [ ( ) ( )]ψ ρ ψ ρ ψ′i i iQ E W . Similarly, by assumption 1-3 and 5 we can show

2 22

, ,

( , ) ( , )( , ) ( , ) < ,sup supψ g ψ g

ρ ψ g ρ ψ gρ ψ g ρ ψ gg g′ ′

Ω Ω Ω Ω

∂ ∂≤ ∞ ′ ′∂ ∂

i ii i i i iE A E A E A

then again by the ULLN,

, =1

( , )1 ( , ) = (1) ( . .)supψ g

ρ ψ gρ ψ gg′

Ω Ω

∂′∂∑

Ni

i ii

A O a sN

Therefore,

0=1

( , )1 ˆ( , ) ( )supψ

ρ ψ gρ ψ g g gg′

Ω

∂−

′∂∑

Ni

i i Ni

AN

. .

0, =1

( , )1 ˆ( , ) 0.supψ g

ρ ψ gρ ψ g g gg′

Ω Ω

∂≤ − →

′∂∑N a s

ii i N

iA

N (29)

It follows (27) - (??) that. .1 ( ) ( ) 0.sup

g

ψ ψΩ

− →a s

NQ QN (30)

Furthermore, 0 0 0( ) = ( ) [ ( ) ( )) ( ( ) ( )]ψ ψ ρ ψ ρ ψ ρ ψ ρ ψ′+ − −i i i i iQ Q E A ,

then by assumption 4, 0( ) ( )ψ ψ≥Q Q and the equality holds if and only if ψ = ψ0. Thus, all conditions of Amemiya [28] Lemma 3 are satisfied and therefore

. .

0ψ ψ→a s

N , as N→∞.

Proof of Theorem 1.2

By assumption 5 and the DCT, the first derivative ( ) /ψ ψ∂ ∂NQ

N = 50 N = 100 Parameter Naive MLE MME Naive MLE MME β0 = 0.00 1.65 (1.69) 0.02 (0.15) 1.61 (1.62) 0.01 (0.08) βx = 2.00 -1.31 (1.32) 0.07 (0.72) -1.32 (1.32) 0.03 (0.12) βz = 1.00 -0.10 (0.22) -0.05 (0.49) -0.11 (0.16) 0.01 (0.22) θ = 0.50 0.64 (1.06) 0.11 (1.09) 0.51 (0.65) 0.05 (0.13)

Table 2: Biases(RMSE) for the parameter estimates in the random intercept Lo-gistic models.

N =100 N =300

Distribution Parameter Naive MLE MME Naive MLE MME

Normal β0 = 0.00 0.11 (0.16) 0.01 (0.09) 0.10 (0.12) 0.01 (0.06)

βx = 2.00 -0.51 (0.51) -0.05 (0.07) -0.51 (0.51) -0.02 (0.05)

θ = 0.25 0.00 (0.18) 0.00 (0.25) 0.00 (0.11) 0.01 (0.17)

ϕ =1 2.99 (3.00) 0.09 (0.46) 3.00 (3.01) 0.06 (0.31)

t(3) β0 = 0.00 0.10 (0.15) 0.00 (0.10) 0.10 (0.12) 0.01 (0.06)

βx = 2.00 -0.50 (0.51) -0.04 (0.07) -0.50 (0.51) -0.02 (0.05)

θ = 0.25 0.01 (0.38) -0.02 (0.29) -0.01 (0.20) -0.02 (0.17)

ϕ =1 2.98 (2.99) 0.05 (0.48) 2.99 (3.00) 0.05 (0.28)

Table 3: Biases(RMSE) for the parameter estimates in the random intercept Pois-son models.

Citation: Li H, Wang L (2012) Consistent Estimation in Generalized Linear Mixed Models with Measurement Error. J Biomet Biostat S7:007. doi:10.4172/2155-6180.S7-007

Page 6 of 7

J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journalMedical statistics: Clinical and Experimental Research

exists and has the first-order Taylor expansion in the open neighborhood

Ωψ0 ∈ Ωψ of ψ0. Since . .

0ψ ψ→a s

N , for sufficiently large N we have 2

00

ˆ( ) ( ) ( ) ˆ= ( ) = 0,ψ ψ ψ ψ ψψ ψ ψ ψ

∂ ∂ ∂+ −

′∂ ∂ ∂ ∂N N N N N

NQ Q Q (31)

where 0 0ˆψ ψ ψ ψ− ≤ −N N . The first and second derivative of ( )ψNQ

in (31) are given in (17) and (18).

Analogous to the proof of Theorem 1.1, by assumption 1 - 5 and Cauchy-Schwartz inequality, we can verify that

2( ) ( ) ( ) <sup sup

ψ ψ

ρ ψ ρ ψ ρ ψψ ψ ψ′ ′

Ω Ω

∂ ∂ ∂≤ ∞

′∂ ∂ ∂i i i

i iE A E A

and ( ( ) / )( ( ) ) < .sup

ψ

ρ ψ ψρ ψψ′

′Ω

∂ ∂ ∂⊗ ∞

′∂i

i ivecE A I

Therefore by the ULLN and Lemma 4 of Amemiya [28], we have 2 . .

0 0 00

( ) ( ) ( ) ( ( ) / )1 ( ( ) ) =2 ψ

ψ ρ ψ ρ ψ ρ ψ ψρ ψψ ψ ψ ψ ψ

′ ′′

∂ ∂ ∂ ∂ ∂ ∂ → + ⊗ ′ ′ ′∂ ∂ ∂ ∂ ∂

a sN i i i

i i iQ vecE A A I D

N

(32)

where Dψ is given in (11) and the second equality holds because

00

( ( ) / )( ( ) ) = 0.ρ ψ ψρ ψψ′

′ ∂ ∂ ∂

⊗ ′∂ i

i ivecE A I

Then by assumption 6 and (32), we rearrange (31) as

1 00

( )1ˆ( ) = (2 )ψψψ ψ

ψ− ∂

− − ∂ N

NQN D

N (33)

For by assumption 5 and DCT, we have the first-order Taylor

expansion of 0( )ψψ

∂∂NQ about γ0:

20 0 0

0 0=1

( ) ( ) ( ) ˆ= 2 ( ) ( ),ψ ρ ψ ψρ ψ g gψ ψ ψ g

′∂ ∂ ∂+ −

′∂ ∂ ∂ ∂∑

NN i N

i ii

Q QA (34)

where 0 0ˆg g g g− ≤ − and

0 02

0

=1 00

( , ) ( , )( ) = 2 .

ˆ( ( , ) / )( ( , ) )

ρ ψ g ρ ψ gψ gψ

ρ ψ g ψψ g ρ ψ gg

′ ′

′′

∂ ∂ ′∂ ∂∂

′ ∂ ∂ ∂∂ ∂ + ⊗ ′∂

∑

i iiN

N

i ii i

AQ

vecA I

Similarly to the derivation of (32), we can show 2 . .

0 0 0( ) ( ) ( )1 = .2 ψg

ψ ρ ψ ρ ψψ g ψ g

′∂ ∂ ∂ → ′ ′∂ ∂ ∂ ∂

a sN i i

iQ E A D

N (35)

Then by (33)-(35), we have

1 1/2 100 0 0

=1

( )ˆ ˆ( ) = ( ) ( )ψ ψ ψgρ ψψ ψ ρ ψ g g

ψ− − −′ ∂

− − + − ∂ ∑

Ni

N i ii

N D N A D D N (36)

Therefore, if Dψγ=0 we can ignore the effect of g and simply treated it as a known constant. If Dψ ≠ 0, we need to make some adjustments to the asymptotic variance of . Since , we have the first-order Taylor expansion in the open neighborhood of γ0

20

0

ˆ ( )( ) ( ) ˆ= ( ) = 0,gg g g gg g g g

∂Ψ∂Ψ ∂ Ψ+ −

′∂ ∂ ∂ ∂

N (37)

By assumption 6, we can have the following representation of g N

1 1/2 1/2 10

=1 =1

( ) ( )ˆ( ) = ( ) = ( ) ,g gg gg g g g

g g− − −′ ′ ∂ ∂

− − − ∂ ∂ ∑ ∑

N Ni i

i ii i

r rN D N r N D r (38)

where Dγ is given in (12). Then plug it back into (36), we have

1 1/2 100 0

=1

( ) ( )ˆ( ) = ( ) ( )ψ ψg gρ ψ gψ ψ ρ ψ g

ψ g− − −′ ′ ∂ ∂

− − + ∂ ∂ ∑

Ni i

N i i ii

rN D N A D D r

Finally, the theorem follows from (31) - (??), CLT and Slutsky's Theorem.

Proof of Theorem 2.1

By assumption 1 and the DCT, , ( )ψN SQ has the first-order Taylor expansion about γ0,

, ,1 ,2=1

( ) = '( ) ( )ψ ρ ψ ρ ψ∑N

N S i i ii

Q A

,2 ,1,1 ,2 0

=1

( , ) ( , )ˆ'( , ) '( , ) ( ),

ρ ψ g ρ ψ gρ ψ g ρ ψ g g g

g g∂ ∂

+ + − ′ ′∂ ∂ ∑

Ni i

i i i i Ni

A A (39)

where 0 0ˆg g ψ ψ− ≤ −N N . Since ρi,1 and ρi,2 are conditionally independent given (Yi,Wi,Vi,Zi,Bi), analogous to the proof of Theorem 1.1, by assumptions 1-3 and Cauchy-Schwartz inequality, we have

,1 ,2| ( ) ( ) | < .sup ρ ψ ρ ψΓ

′ ∞ i i iE A

Hence by the ULLN, . .

,1 ,2 ,1 ,2=1

1 ( ) ( ) '( ) ( )ρ ψ ρ ψ ρ ψ ρ ψ′ →∑a sN

i i i i i iiA E W

N

uniformly in ψ ∈ Г, where

,1 ,2 ,1'( ) ( ) = [ ( '( ) | , , , , )ρ ψ ρ ψ ρ ψi i i i i i i i iE W E E Y W V Z B

,2( '( ) | , , , , )] = ( ).ρ ψ ψi i i i i i iW E Y W V Z B Q

Similar to proof of Theorem 1.1, we can show that ,1

,2 0=1

( , )1 ˆ'( , ) ( )supρ ψ g

ρ ψ g g ggΓ

∂−

′∂∑

Ni

i i Ni

AN

. .,1

,2 0( , ) =1

( , )1 ˆ'( , ) 0.supρ ψ g

ρ ψ g g ggΓ ϒ

∂≤ − →

′∂∑

N a si

i i Ni

AN

(40)

It then follows that . .

,1 ( ) ( ) 0.sup ψ ψ

Γ− →

a s

N SQ QN (41)

It has been proved previously that ( )ψQ attains a unique minimum

at ψ0 ∈ Г. Therefore, by Lemma 3 of Amemiya (1973), . .

, 0ψ ψ→a s

N S , as

N → ∞.

Proof of Theorem 2.2

For sufficiently large N, by assumption 5 we have the first-order Taylor expansion of , ( ) /ψ ψ∂ ∂N SQ about ψ0:

2, 0 ,

, 0

( ) ( )ˆ( ) = 0,

ψ ψψ ψ

ψ ψ ψ∂ ∂

+ −′∂ ∂ ∂

N S N SN S

Q Q(42)

where 0 , 0ˆψ ψ ψ ψ− ≤ − N S and the first and second derives are given by

, ,1 ,2,2 ,1

=1

ˆ ˆ( ) '( ) '( )ˆ ˆ= ( ) ( )

ψ ρ ψ ρ ψρ ψ ρ ψ

ψ ψ ψ∂ ∂ ∂

+ ∂ ∂ ∂ ∑

NN S i i

i i i ii

QA A

and2

, ,1 ,2 ,1,2

=1

ˆ ˆ ˆ( ) '( ) ( ) ( '( ) / )ˆ= ( '( ) )

ψ ρ ψ ρ ψ ρ ψ ψρ ψ

ψ ψ ψ ψ ψ∂ ∂ ∂ ∂ ∂ ∂

+ ⊗ ′ ′ ′∂ ∂ ∂ ∂ ∂ ∑

NN S i i i

i i ii

Q vecA A I

,2 ,1 ,2,1

=1

ˆ ˆ ˆ'( ) ( ) ( '( ) / )ˆ( '( ) ) .

ρ ψ ρ ψ ρ ψ ψρ ψ

ψ ψ ψ∂ ∂ ∂ ∂ ∂

+ + ⊗ ′ ′∂ ∂ ∂ ∑

Ni i i

i i ii

vecA A I

Similar to the derivation of (32), we can show 2

, ( )1 ψψ ψ

∂′∂ ∂

N SQN

converges to

,1 0 ,2 0 ,1 0,2 0

'( ) ( ) ( '( ) / )( '( ) )i i i

i i i

vecE A A I

ρ ψ ρ ψ ρ ψ ψρ ψ

ψ ψ ψ∂ ∂ ∂ ∂ ∂

+ ⊗ ′ ′∂ ∂ ∂

Citation: Li H, Wang L (2012) Consistent Estimation in Generalized Linear Mixed Models with Measurement Error. J Biomet Biostat S7:007. doi:10.4172/2155-6180.S7-007

Page 7 of 7

J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journalMedical statistics: Clinical and Experimental Research

,2 0 ,1 0 ,2 0,1 0

'( ) ( ) ( '( ) / )( '( ) ) ,

ρ ψ ρ ψ ρ ψ ψρ ψ

ψ ψ ψ∂ ∂ ∂ ∂ ∂

+ + ⊗ ′ ′∂ ∂ ∂ i i i

i i i

vecE A A I

uniformly for all ψ ∈ Г. Since

,1 0 ,2 0 0 0'( ) ( ) '( ) ( )= = ψ

ρ ψ ρ ψ ρ ψ ρ ψψ ψ ψ ψ

∂ ∂ ∂ ∂ ′ ′∂ ∂ ∂ ∂

i i i ii iE A E A D

and,1 0

,2 0

( '( ) / )( '( ) ) = 0,

ρ ψ ψρ ψ

ψ∂ ∂ ∂

⊗ ′∂ i

i i

vecE A I

we have 2 . .

, ( )1 2 .ψ

ψ ψ∂

→′∂ ∂

a sN SQ

DN

(43)

Again, , 0( )ψ ψ∂ ∂N SQ has the first-order Taylor expansion about γ0: 2

, 0 ,1 0 ,2 0 , 0,2 0 ,1 0 0

=1

( ) ( ) ( ) ( )ˆ= ( ) ( ) ( ).

ψ ρ ψ ρ ψ ψρ ψ ρ ψ g g

ψ ψ ψ ψ g′ ′∂ ∂ ∂ ∂ ′ ′+ + − ′∂ ∂ ∂ ∂ ∂

∑N

N S i i N Si i i i N

i

Q QA A

(44)Finally, Analogous to the proof of Theorem 1.2, the results follows

from (38), (42)-(44), CLT and Slutsky's Theorem.

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This article was originally published in a special issue,Medical statistics: Clinical and experimental researchhandledbyEditor(s).Dr.HerbertPang,DukeUniversity,USA.